Bump map connected to Glossy shader Normal input causes line artifact in render
In short, adding all those values together is making bump values that are larger than the reasonable range.

The add nodes don't do anything magical. They are doing exactly what it sounds like ;-) They're adding numbers together. In the case of bump, full normal distortion is 1 (tallest value), no normal distortion is 0 (shallowest value).

50% grey plus 50% grey is 100% white, which is full bump distortion. 75% grey plus 75% grey is.... a color that Blender doesn't know what to do with... which is a bump distortion that Blender doesn't know what to do with.

Those add nodes shouldn't have anything to do with color. You can't necessarily expect that the color you want is also exactly the bump value that you want. You should manipulate the color of your shader seperately from the bump of your shader, even if you have to input the same image twice.

I hope this helps!